I’ve finally read through all of Lockhart’s Lament, which was referenced in an earlier Decryptions post. Within, Lockhart tears apart what most of the population thinks of as “math.” He ponders, “I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.” Instead, he stirs up beautiful descriptions of the adventure and fantasy of pondering an abstract idea, discovering the unexpected, finding new riddles, symmetries, understanding. I’m fortunate to belong to a small group of friends that shares similar delights, enjoying the adventure of exploration and discovery.
Ok, so most of the people we see day-to-day absolutely do not share in this. Mention to a typical adult the idea of helping a 6th grader with math and they get kind of an uncertain, unhappy look, like finding a thick black hair in their first bite of pizza. Lockhart is “complaining about the complete absence of art and invention, history and philosophy, context and perspective from the mathematics curriculum. That doesn’t mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both.”
Unfortunately, the article concludes without much hope of drastic change to our “lecture, test, repeat” standards of the day. So, I’ve been observing the way my elementary schoolers are learning and pondering how I learned. I didn’t have nearly as much fun as the students described in Lockhart’s classroom, but I don’t think I completely missed the adventure, either. I remember often relishing that feeling of finally getting my head around the underlying principles of the topic of the week. Perhaps teachers should help students enjoy this level of “getting it” more than they do. Give them a chance to play around before rushing off to the next standard test topic. In the Dreyfus Model of Skill Acquisition, the novice starts with “rigid adherence to taught rules or plans.” The learner goes through “Competence, Proficient, Expertise, and Mastery.” Perhaps this applies iteratively to each domain of math one is challenged by. First one must discover the shape of a thing, or its “nature,” before questions and intuitions can be applied. I see mastery in my kids’ learning regularly. Three months ago, my 3rd grader was stumped by adding and subtracting hours and minutes from a given time. Now it’s natural – and fun! I believe he’s achieved a deeper understanding than the “trained chimpanzee” Lockhart references.
So, how to allow one’s own children to experience that thrill of discovery and to know that they are fully capable to use their creativity in unrestricted ways? Take Lockhart’s advice and “Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever.” Come into the classroom and do an hour presentation on your field of interest. In what ways have you applied your creativity lately? Engage your children at home in exciting ways. I’ll frequently find some article about engineering or science and discuss it with them in unabashed wonder. We’ll be opening an affordable 3D printer in a couple of days. We’ve already started using Google Sketchup to get a taste of what it is to bring an idea into existence.
Decryptions 
Lockhart says “Play games!” and I agree, but I worry that people will take that to mean no more doing math problem sets. In competitive sports the idea of drills to ground players in fundamentals doesn’t seem to to the same opposition that doing math exercises to teach fundamentals does.
I’m not suggesting that the current way math is taught is right, but that a deeper look at the problem will have to consider that there are some things which are just hard to do. How do they do it in sports? Part of it seems to be linking those skills to concrete problems (no, ” Bob is 4 years older than Alice and Alice is half as old as Dave” is not going to cut it). Part of it is that drills lead to something students want to do. Math seems to be all about the drills and nothing else.
I have to admit though I’m not sure what activities would work, especially for younger students.
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